Previous |  Up |  Next

# Article

Full entry | Fulltext not available (moving wall 24 months)
Keywords:
distance spectrum; distance characteristic polynomial; $D$-spectrum determined by its $D$-spectrum
Summary:
Let $G$ be a connected graph with vertex set $V(G)=\{v_{1},v_{2},\ldots ,v_{n}\}$. The distance matrix $D(G)=(d_{ij})_{n\times n}$ is the matrix indexed by the vertices of\/ $G$, where $d_{ij}$ denotes the distance between the vertices $v_{i}$ and $v_{j}$. Suppose that $\lambda _{1}(D)\geq \lambda _{2}(D)\geq \nobreak \cdots \geq \lambda _{n}(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its \hbox {$D$-spectrum} if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra.
References:
[1] Cioabă, S. M., Haemers, W. H., Vermette, J. R., Wong, W.: The graphs with all but two eigenvalues equal to $\pm 1$. J. Algebra Comb. 41 (2015), 887-897. DOI 10.1007/s10801-014-0557-y | MR 3328184 | Zbl 1317.05111
[2] Cvetković, D. M., Doob, M., Sachs, H.: Spectra of Graphs. Theory and Applications. J. A. Barth Verlag, Heidelberg (1995). DOI 10.1002/zamm.19960760305 | MR 1324340 | Zbl 0824.05046
[3] Günthard, H. H., Primas, H.: Zusammenhang von Graphentheorie und MO-Theorie von Molekeln mit systemen konjugierter Bindungen. Helv. Chim. Acta 39 (1956), 1645-1653 German. DOI 10.1002/hlca.19560390623
[4] Jin, Y.-L., Zhang, X.-D.: Complete multipartite graphs are determined by their distance spectra. Linear Algebra Appl. 448 (2014), 285-291. DOI 10.1016/j.laa.2014.01.029 | MR 3182986 | Zbl 1285.05114
[5] Lu, L., Huang, Q. X., Huang, X. Y.: The graphs with exactly two distance eigenvalues different from $-1$ and $-3$. J. Algebr. Comb. 45 (2017), 629-647. DOI 10.1007/s10801-016-0718-2 | MR 3604069 | Zbl 1358.05176
[6] Lin, H. Q.: On the least distance eigenvalue and its applications on the distance spread. Discrete Math. 338 (2015), 868-874. DOI 10.1016/j.disc.2015.01.00610.1016/j.disc.2015.01.006 | MR 3318625 | Zbl 1371.05064
[7] Lin, H. Q., Hong, Y., Wang, J. F., Shu, J. L.: On the distance spectrum of graphs. Linear Algebra Appl. 439 (2013), 1662-1669. DOI 10.1016/j.laa.2013.04.019 | MR 3073894 | Zbl 1282.05132
[8] Lin, H. Q., Zhai, M. Q., Gong, S. C.: On graphs with at least three distance eigenvalues less than $-1$. Linear Algebra Appl. 458 (2014), 548-558. DOI 10.1016/j.laa.2014.06.040 | MR 3231834 | Zbl 1296.05123
[9] Liu, R. F., Xue, J., Guo, L. T.: On the second largest distance eigenvalue of a graph. Linear Multilinear Algebra 65 (2017), 1011-1021. DOI 1080/03081087.2016.1221376 | MR 3610302 | Zbl 1360.05099
[10] Dam, E. R. van, Haemers, W. H.: Which graphs are determined by their spectrum?. Linear Algebra Appl. 373 (2003), 241-272. DOI 10.1016/S0024-3795(03)00483-X | MR 2022290 | Zbl 1026.05079
[11] Dam, E. R. van, Haemers, W. H.: Developments on spectral characterizations of graphs. Discrete Math. 309 (2009), 576-586. DOI 10.1016/j.disc.2008.08.019 | MR 2499010 | Zbl 1205.05156
[12] Xue, J., Liu, R. F., Jia, H. C.: On the distance spectrum of trees. Filomat 30 (2016), 1559-1565. DOI 10.2298/FIL1606559X | MR 3530101 | Zbl 06749814

Partner of